Question

which function is represented by the graph below?
options:
$f(x) = -\ln(2x)$
$f(x) = -2e^x$
$f(x) = -2\ln(x)$
$f(x) = e^{-2x}$

Explanation:

Step1: Identify graph type

The graph has a vertical asymptote at \(x=0\) (right side only), so it's a logarithmic function (exponential functions have no vertical asymptotes). Eliminate exponential options B and D.

Step2: Test \(x=1\)

For \(f(x)=-\ln(2x)\): \(f(1)=-\ln(2)\approx-0.693\).
For \(f(x)=-2\ln(x)\): \(f(1)=-2\ln(1)=0\).
The graph passes through \(x=1\) at \(y\approx-0.693\), matching \(f(x)=-\ln(2x)\).

Answer:

A. \(f(x) = -\ln(2x)\)